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C7×D15⋊D85 #49087987

order35700
automorphism?
homologyAbelianization (H1) = C22×C7 · Schur multiplier (H2) = C2

Aliases

direct productC7×D15⋊D85
unambiguous semidirect productD15⋊(C7×D85)

Splittings

Fitting, socle, HallC3×C52×C7×C17 ⋊ C22
derived, HallC3×C52×C17 ⋊ C22×C7
HallC52×C7×C17 ⋊ C2×D3
HallC3×C52×C7 ⋊ C2×D17
HallC52×C17 ⋊ C2×D3×C7
HallC3×C7×C17 ⋊ D52
HallC52×C7 ⋊ D3×D17
HallC7×C17 ⋊ D15⋊D5
HallC3×C52 ⋊ C2×C7×D17
HallC3×C17 ⋊ C7×D52
SylowC52 ⋊ D3×C7×D17
HallC3×C7 ⋊ D5×D85
SylowC17 ⋊ C7×D15⋊D5
SylowC3 ⋊ C7×D5×D85

Subgroups

subgroupG/H
center, hypercenterC7D15⋊D85
commutator, cosocle, hypocenterC3×C52×C17C22×C7
FrattiniC1G
Fitting, socleC3×C52×C7×C17C22
LCSG ▷ C3×C52×C17
UCSC1 ◁ C7

192 conjugacy classes of subgroups; only the 50 normal subgroups are shown.

normal subgroups only

This group has 192 conjugacy classes of subgroups — too many to draw. Only its 50 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

Full subgroup data →

Presentation

minimal degreeμ(G) = 37
transitive degreeμt(G) = 3570

A minimal faithful permutation representation — 2 generators on 37 points:

(1,2,3,4,5,6,7)(9,10)(11,12)(13,14)(16,17,19,20,18)(21,22,24,26,28,30,32,34,36,37,35,33,31,29,27,25,23);(8,9,10)(12,13)(14,15)(17,18)(19,20)(22,23)(24,25)(26,27)(28,29)(30,31)(32,33)(34,35)(36,37)

Conjugacy classes & character table

2730 conjugacy classes. The character table is too large to display.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(35700, 461);